Make Sense of Exponent Properties

How to Teach the Exponent Properties

Students often find exponent properties confusing, especially since addition is used to simplify expressions like $x^5x^3$ despite no visible plus sign. Instead of just telling students the rules, which is ineffective for those who struggle with memorization, a better approach is a four-step process:

1. Make sense of the expression
2. Model the expression
3. Simplify using the model
4. Notice the pattern

This method helps students discover and understand exponent properties conceptually.

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For the expression $x^5x^3$:

1. Make Sense of the Expression
I would start by asking my students, "What does this expression mean?" Hopefully, they could make sense that it is the product of $x$ multiplied by itself 5 times and $x$ multiplied by itself 3 times.

2. Modeling the Expression
I would then have them write out the meaning of what they explained in the first step: $(x.x.x.x.x)(x.x.x)$.

3. Simplifying the Expression
We would then talk about how $(x.x.x.x.x)(x.x.x)$ is the same as $x^8$.

4. Noticing Patterns
After doing a couple of similar problems following this method, I would then ask students to look for a pattern. Hopefully, they would notice that the exponent in the simplified expression is the sum of the exponents with the same bases.

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For the expression $x^5{\mathrm{/x}}^3$:

1. Make Sense of the Expression
Ask students what the expression means, helping them recognize it represents division.

2. Modeling the Expression
Write it out as $(x.x.x.x.x)/(x.x.x)$ to represent what they described.

3. Simplifying the Expression
Review that $x/x = 1$, just like $2/2 = 1$ or $5/5 = 1$. After canceling, the expression simplifies to $x^2$.

4.Noticing Patterns
Repeat with similar problems and ask students to find a shortcut. They should see that the exponent in the simplified expression is the difference of the exponents in the original expression.

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I use this method for every exponent rule. For negative exponents, I create a table with positive exponents and guide students to extend the pattern. This approach helps students truly understand exponent properties rather than relying on memorization. If they forget a rule, they can always model and simplify instead. For complex expressions like $x^{50}x^{30}$, rather than modeling all the $x$'s, I give them a simpler expression to work through first, helping them recognize the pattern and apply it to the larger problem.

Exponent Properties Guided Notes for YOU

If you are looking for some guided notes on exponent properties I have taken then time to create some.  Students will discover all the rules through this method and apply their learning on expressions.  

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Save these tips and ideas to your favorite classroom Pinterest board. Come back and reference them for ideas on teaching the exponent properties.